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The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula. We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general. [ABSTRACT FROM AUTHOR]

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements that violate the shape regularity or maximum angle condition are covered virtually by simplices that satisfy the minimum angle condition. A numerical experiment illustrates the theoretical result. [ABSTRACT FROM AUTHOR]

Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study n s -step interpolatory M -subdivision schemes and their interpolating M -refinable functions with n s ∈ N ∪ { ∞ } and a dilation factor M ∈ N \ { 1 } . We completely characterize C m -convergence and smoothness of n s -step interpolatory subdivision schemes and their interpolating M -refinable functions in terms of their masks. Inspired by n s -step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their C m -convergence and smoothness properties using only their masks. Moreover, combining n s -step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain r n s -step interpolatory subdivision schemes. Examples and construction procedures of convergent n s -step interpolatory M -subdivision schemes are provided to illustrate our results with dilation factors M = 2 , 3 , 4 . In addition, for the dyadic dilation M = 2 and r = 2 , 3 , using r masks with only two-ring stencils, we provide examples of C r -convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes. [ABSTRACT FROM AUTHOR]

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures. [ABSTRACT FROM AUTHOR]

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures. [ABSTRACT FROM AUTHOR]

Spectral collocation method, named linear barycentric rational interpolation collocation method (LBRICM), for convection-diffusion (C-D) equation with constant coefficient is considered. We change the discrete linear equations into the matrix equation. Different from the classical methods to solve the C-D equation, we solve the C-D equation with the time variable and space variable obtained at the same time. Furthermore, the convergence rate of the C-D equation by LBRICM is proved. Numerical examples are presented to test our analysis.

We present a derivative-free separable quadratic modeling and cubic regularization technique for solving smooth unconstrained minimization problems. The derivative-free approach is mainly concerned with building a quadratic model that could be generated by numerical interpolation or using a minimum Frobenius norm approach, when the number of points available does not allow to build a complete quadratic model. This model plays a key role to generate an approximated gradient vector and Hessian matrix of the objective function at every iteration. We add a specialized cubic regularization strategy to minimize the quadratic model at each iteration, that makes use of separability. We discuss convergence results, including worst case complexity, of the proposed schemes to first-order stationary points. Some preliminary numerical results are presented to illustrate the robustness of the specialized separable cubic algorithm. [ABSTRACT FROM AUTHOR]

Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in the MD displacement data sets. Inspired by the weighted essentially non-oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as the piecewise smooth solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization-based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the first phase, we identify the optimal nonlocal kernel function from the data sets without material damage to capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with MD. We illustrate the efficacy of the proposed approach with several numerical tests for the dynamic crack propagation problem in a single-layer graphene. Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable of modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training. [ABSTRACT FROM AUTHOR]

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement. [ABSTRACT FROM AUTHOR]

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices. [ABSTRACT FROM AUTHOR]

We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average — a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach. [ABSTRACT FROM AUTHOR]

Numerical analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability of the evaluation creeps in through the back door. This paper shows for a large class of numerical methods that such a Trade-off Principle between error and evaluation stability is unavoidable. It is an instance of a no free lunch theorem. Here, evaluation is the mathematical map that takes input data to output data. This is independent from the numerical routine that calculates the output. Therefore, evaluation stability is different from computational stability. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. The trade-off principle bounds the product of two terms from below. The first is related to errors, and the second turns out to be related to evaluation instability. Under certain conditions satisfied for splines and kernel-based interpolation, both can be minimized. Then the lower bound is attained, and the error term is the inverse of the instability term. As a byproduct, it is shown that Kansa’s Unsymmetric Collocation Method sacrifices accuracy for improved evaluation stability, when compared to symmetric collocation. [ABSTRACT FROM AUTHOR]

In this digital, internet-based world, it is not new to face cyber attacks from time to time. A number of heavy viruses have been made by hackers, and they have successfully given big losses to our systems. In the family of these viruses, the Stuxnet virus is a well-known name. Stuxnet is a very dangerous virus that probably targets the control systems of our industry. The main source of this virus can be an infected USB drive or flash drive. In this research paper, we study a mathematical model to define the dynamical structure or the effects of the Stuxnet virus on our computer systems. To study the given dynamics, we use a modified version of the Caputo-type fractional derivative, which can be used as an old Caputo derivative by fixing some slight changes, which is an advantage of this study. We demonstrate that the given fractional Caputo-type dynamical model has a unique solution using fixed point theory. We derive the solution of the proposed non-linear non-classical model with the application of a recent version of the Predictor–Corrector scheme. We analyze various graphs at different values of the arrival rate of new computers, damage rate, virus transmission rate, and natural removal rate. In the graphical interpretations, we verify the values of fractional orders and simulate 2-D and 3-D graphics to understand the dynamics clearly. The major novelty of this study is that we formulate the optimal control problem and its important consequences both theoretically and mathematically, which can be further extended graphically. The main contribution of this research work is to provide some novel results on the Stuxnet virus dynamics and explore the uses of fractional derivatives in computer science. The given methodology is effective, fully novel, and very easy to understand.

Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

In this paper, we address the problem of constructing C 2 cubic spline functions on a given arbitrary triangulation T . To this end, we endow every triangle of T with a Wang–Shi macro-structure. The C 2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of C 2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for C 2 joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang–Shi macro-structure is transparent to the user. Stable global bases for the full space of C 2 cubics on the Wang–Shi refined triangulation T are deduced from the local simplex spline basis by extending the concept of minimal determining sets. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

The intersection of surfaces is a fundamental process in computational geometry and computer-aided design applications to build and interrogate complex shapes in the computer. This paper presents a novel and simple dual quaternion-based osculating circle DQOC algorithm to find the intersection curve of two regular surfaces based on the osculating circle concept and dual quaternions. Additionally, we expressed the natural equations of the intersection curve. We have also demonstrated the superiority of our method through numerical examples.

Facial expression generation has always been an intriguing task for scientists and researchers all over the globe. In this context, we present our novel approach for generating videos of the six basic facial expressions. Starting from a single neutral facial image and a label indicating the desired facial expression, we aim to synthesize a video of the given identity performing the specified facial expression. Our approach, referred to as FEV-GAN (Facial Expression Video GAN), is based on Spatio-temporal Convolutional GANs, that are known to model both content and motion in the same network. Previous methods based on such a network have shown a good ability to generate coherent videos with smooth temporal evolution. However, they still suffer from low image quality and low identity preservation capability. In this work, we address this problem by using a generator composed of two image encoders. The first one is pre-trained for facial identity feature extraction and the second for spatial feature extraction. We have qualitatively and quantitatively evaluated our model on two international facial expression benchmark databases: MUG and Oulu-CASIA NIR&VIS. The experimental results analysis demonstrates the effectiveness of our approach in generating videos of the six basic facial expressions while preserving the input identity. The analysis also proves that the use of both identity and spatial features enhances the decoder ability to better preserve the identity and generate high-quality videos. The code and the pre-trained model will soon be made publicly available.

Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, and 3D object analysis. This paper studies the problem of learning real-valued functions on manifolds through filtered hyperinterpolation of input–output data pairs where the inputs may be sampled deterministically or at random and the outputs may be clean or noisy. Motivated by the problem of handling large data sets, it presents a parallel data processing approach which distributes the data-fitting task among multiple servers and synthesizes the fitted sub-models into a global estimator. We prove quantitative relations between the approximation quality of the learned function over the entire manifold, the type of target function, the number of servers, and the number and type of available samples. We obtain the approximation rates of convergence for distributed and non-distributed approaches. For the non-distributed case, the approximation order is optimal. [ABSTRACT FROM AUTHOR]

The Floater--Hormann family of rational interpolants do not have spuriouspoles or unattainable points, are efficient to calculate, and have arbitrarilyhigh approximation orders. One concern when using them is that theamplification of rounding errors increases with approximation order, and canmake balancing the interpolation error and rounding error difficult. Thisarticle proposes to modify the Floater--Hormann interpolants by includingadditional local polynomial interpolants at the ends of the interval. Thisappears to improve the conditioning of the interpolants and allow higherapproximation orders to be used in practice.

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted α, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter α, at an increased cost that scales as log α . Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible. [ABSTRACT FROM AUTHOR]

A cubic H-Bézier geometric approximation scheme with one free parameter τis introduced for parabolic arcs. It has four control points. The control points are computed by matching the end points and end unit tangents of the two curves and provide two more free parameters. These free parameters are computed by curvature continuity constraints. Optimal value of τ is obtained by minimizing the maximum value of the distance between the cubic H-Bézier curve and the concerned parabolic arc. The method is illustrated using different numerical examples which show that the proposed method is economical, reliable and efficient.

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes. [ABSTRACT FROM AUTHOR]

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study. [ABSTRACT FROM AUTHOR]

In this article, the optimal control for nonlinear SIRC model is studied in fractional order using the Caputo fractional derivative. Graph signal flow is given of the model and simulated by Simulink/Matlab which helps in discussing the topological structure of the model. Dynamics of the system versus certain parameters are studied via bifurcation diagrams, Lyapunov exponents and Poincare maps. The existence of a uniformly stable solution is proved after control. The obtained results display the activeness and suitability of the Mittag Generalized-Leffler function method (MGLFM). The approximate solution of the fractional order SIRC model using MGLFM is explained by giving the figures of solutions before and after control. Also, we plot the 3D relationships with different alpha (fractional order) which display the originality and suitability of the results.

A particle-cluster treecode based on barycentric Hermite interpolation is presented for fast summation of electrostatic particle interactions in 3D. The interpolation nodes are Chebyshev points of the 2nd kind in each cluster. It is noted that barycentric Hermite interpolation is scale-invariant in a certain sense that promotes the treecode’s efficiency. Numerical results for the Coulomb and screened Coulomb potentials show that the treecode run time scales like O(N log N), where N is the number of particles in the system. The advantage of the barycentric Hermite treecode is demonstrated in comparison with treecodes based on Taylor approximation and barycentric Lagrange interpolation.